Herz-Morrey-Hardy Spaces with Variable Exponents and Their Applications

The Herz spaces go back to Beurling and Herz; see [1, 2]. Firstly, they attracted a lot of authors’ attention because they could be used to characterize Fourier multipliers for Hardy spaces; see [3].Then, in 1989 Chen and Lau in [4] andGarćıaCuerva in [5] introduced now called nonhomogeuous Herz type Hardy spaces. They found that these Herz type Hardy spaces have a decomposition via central atoms. After that, Lu et al. considered homogeuous Herz type Hardy spaces and also obtained a central atomic decomposition for them. Since then Herz type spaces have been studied extensively; see monograph [6] for details. Meanwhile, in the last three decades, the interest of the study for variable exponent spaces has been increasing year by year. Variable exponent spaces have many applications: in electrorheological fluid [7], in differential equations [8] and references therein, and in image restoration [9–11], for instance. Indeed, many spaces with variable exponents appeared, such as: Lebesgue spaces, Sobolev spaces and Bessel potential spaces with variable exponent, Besov and Triebel-Lizorkin spaces with variable exponents, Morrey spaces with variable exponents, Campanato spaces with variable exponent, and Hardy spaces with variable exponent; see [12–23] and references therein. Moreover, the atomic, molecular, and wavelet decompositions of variable exponent Besov and Triebel-Lizorkin spaces were given in [13, 14, 20, 21, 24]. The duality and reflexivity of spaces Bs p(⋅),q and Fs p(⋅),q were discussed in [25]. The atomic and molecular decompositions of Hardy spaces with variable exponent and their applications for the boundedness of singular integral operators were obtained in [22, 26]. Recently, as a generalization of Lebesgue spaces with variable exponent, Herz spaces with variable exponents are introduced. In fact, in 2010 Izuki proved the boundedness of sublinear operators on Herz space with variable exponents


Introduction
The Herz spaces go back to Beurling and Herz; see [1,2].Firstly, they attracted a lot of authors' attention because they could be used to characterize Fourier multipliers for Hardy spaces; see [3].Then, in 1989 Chen and Lau in [4] and García-Cuerva in [5] introduced now called nonhomogeuous Herz type Hardy spaces.They found that these Herz type Hardy spaces have a decomposition via central atoms.After that, Lu et al. considered homogeuous Herz type Hardy spaces and also obtained a central atomic decomposition for them.Since then Herz type spaces have been studied extensively; see monograph [6] for details.Meanwhile, in the last three decades, the interest of the study for variable exponent spaces has been increasing year by year.Variable exponent spaces have many applications: in electrorheological fluid [7], in differential equations [8] and references therein, and in image restoration [9][10][11], for instance.Indeed, many spaces with variable exponents appeared, such as: Lebesgue spaces, Sobolev spaces and Bessel potential spaces with variable exponent, Besov and Triebel-Lizorkin spaces with variable exponents, Morrey spaces with variable exponents, Campanato spaces with variable exponent, and Hardy spaces with variable exponent; see [12][13][14][15][16][17][18][19][20][21][22][23] and references therein.Moreover, the atomic, molecular, and wavelet decompositions of variable exponent Besov and Triebel-Lizorkin spaces were given in [13,14,20,21,24].The duality and reflexivity of spaces   (⋅), and   (⋅), were discussed in [25].The atomic and molecular decompositions of Hardy spaces with variable exponent and their applications for the boundedness of singular integral operators were obtained in [22,26].
Recently, as a generalization of Lebesgue spaces with variable exponent, Herz spaces with variable exponents are introduced.In fact, in 2010 Here there is the usual modification when  = ∞.
Lemma 8 is the generalization of Herz spaces with variable exponents in [28].For a proof, see [33].

The Atomic Characterization
In this section, we will introduce Herz-Morrey-Hardy spaces with variable exponents  K(⋅), (⋅), and  (⋅), (⋅), .To do this, we need to recall some notations.S(R  ) denotes the Schwartz space of all rapidly decreasing infinitely differentiable functions on R  , and S  (R  ) denotes the dual space of S(R  ).Let    be the grand maximal function of  defined by where A  := { ∈ S(R  ) : sup ||,||⩽,∀∈R  |    ()| ⩽ 1} and  >  + 1 and  *  is the nontangential maximal operator defined by The grand maximal operator   was firstly introduced by Fefferman and Stein in [44] to study classical Hardy spaces.For classical Hardy spaces, one can also see [45][46][47].Nakai and Sawano generalized them to variable exponent case in [22].( Remark 10.If (⋅) ≡  and  = 0, these spaces were considered by Wang and Liu in [31].If (⋅) and (⋅) are constant and  = 0, these are the classical Herz type Hardy spaces; see [6].
where the infimum is taken over all above decompositions of .
(ii)  ∈ where the infimum is taken over all above decompositions of .
It is easy to see that Thus, we deduce Similar to the method of [48], let We write where Journal of Function Spaces and b is a constant which will be chosen later.Note that By a computation we have Since it follows that Take b = .It is easy to show that each where  is a constant independent of , , , and .Moreover, we have Using the same argument as before for , we obtain Therefore, Thus, we obtain that where each where  is independent of  and .Since by the Banach-Alaoglu theorem we obtain a subsequence { another application of the Banach-Alaoglu theorem yields a subsequence { Similarly, there exists a subsequence { Repeating the above procedure for each  ∈ Z, we can find a subsequence { which is a central ((⋅), (⋅))-atom supported on  +2 .By using the diagonal method we obtain a subsequence { ] } of N 0 such that, for each  ∈ Z, lim ] → ∞  ( ] )  =   in the weak * topology of  (⋅) and therefore in S  (R  ).Now we only need to prove that  = ∑ ∞ =−∞     in the sense of S  (R  ).For each  ∈ S(R  ), note that supp Using the same argument in [48], we have If  > 0, let  0 ∈ N 0 such that min{ 0 + (0) − ,  0 +  ∞ − } > 0; then by Lemmas 4 and 3 again we have Let Then which implies that This establishes the identity we wanted.
To prove the sufficiency, for convenience, we denote sup where Now we have Therefore, we have To proceed, we consider them into two cases 0 <  ⩽ 1 and 1 <  < ∞.
If 0 <  ⩽ 1, Then we turn to estimate : Third, we estimate : If 1 <  < ∞, we have Second, we estimate .As the same argument before, we obtain that Third, we estimate .We have Thus, we finish the proof of Theorem 13.

Applications
As an application of the atomic decompositions, we will prove the following result.